vd rf lecture

7/27/2019 Vd Rf Lecture

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Resistance forms,quasisymmetric maps and heatkernel estimates

Jun Kigami

Graduate School of Informatics

Kyoto University

Kyoto 606-8501, Japan

e-mail:[email protected] lecture note is available at

http://www-an.acs.i.kyoto-u.ac.jp/kigami/preprints.html

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1 Introduction

(X,d,): a metric measure spaceX: a set, d: a metric on X, : a Borel regular measure on (X, d)

A heat equation on X:u

t= Lu, L is a Laplacian on X

Heat kernel: p(t,x,y), t > 0, x , y X.

u(t, x)

||X

p(t,x,y)u0(y)(dy)||

Ex(u0(Xt))

Transition density: p(t,x,y)

({Xt}t>0, {Px}xX): a Markov process (Hunt process) on X

A regular Dirichlet form (E, F): a quadratic form on L2(X, ) with theMarkov property

E(u, v) = X

u(Lv)d

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Heat kernel estimates:

(1) Brownian motion on Rn the heat equation: ut

= cn

i=1

2

uxi2

Gaussian : p(t,x,y) =c1

tn/2exp

c2 |x y|

2

t

)

(2) Riemannian manifold: Li-Yau(1986)(X, d): complete Riemannan manifold with the Ricci curvature 0

p(t,x,y) c1Vd(x, t1/2)

exp c2d(x, y)2

t ),where Vd(x, r): the volume of a Ball = (Bd(x, r)),Bd(x, r) = {y|d(x, y) < r}.

(3) Brownian motions on Fractals:Sierpinski gasket (Barlow-Perkins), Sierpinski carpet (Barlow-Bass)

sub-Gaussian : p(t,x,y) c1t/

exp

c2

d(x, y)

t

1/(1))

> 2: the walk dimension, : the Hausdorff dimension

(4) the Li-Yau type sub-Gaussian (LY):

p(t,x,y) c1Vd(x, t1/)

exp

c2

d(x, y)

t

1/(1))

General desirable estimate for diffusion processes

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(5) -stable process on Rn:

(0, 2)

Jump process (Pathes of the process are not continuous.)

E()(u, u) =Rn

Rn

|u(x) u(y)|2|x y|n+ dxdy =

Rn

u(x)(

()/2u(x)dxLaplacian L = ()/2: not a local operator

p(t,x,y) min{

tn/,t

|x y|n+}

Convension: f, g : X [0, ). f g def

c1, c2 > 0 such that

c1f(x) g(x) c2f(x)

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Aim of Study 1: Intrinsic meatic

The original metric is not always the best.

Good heat kernel esimate may not always hold under the original metricd. There may exist a metric which is suitable for describing asymptoticbehaviors of the heat kernel.

When and How can we find such a metric?

Aim of Study 2: Regulation of JumpsIf the process is not diffusion, the jumps may cause troubles to describe

asymptotic behaviors.

How can we regulate Jumps?

We will study those problems in the case ofHunt processes associated with resistance forms.

strongly recurrent Hunt processCapacity of a point is positive

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Definition of resistance forms

Definition 1.1. X: a set.(E, F) is called a resistance form on X

def(RF1) through (RF5) hold.

(RF1) F is a linear subspace of (X), 1 F,E: F F R, non-negative symmetricE(u, u) = 0 if and only if u is constant on X.(RF2) Let be an equivalent relation on Fdefined by u v if and only ifu v is constant on X. Then (F/, E) is a Hilbert space.(RF3) x = y u Fsuch that u(x) = u(y).(RF4) For any x, y X,

R(E,F)(x, y) = sup |u(x) u(y)|2

E(u, u) : u F, E(u, u) > 0

< +

(RF5) Markov property: Define u by

u(p) =

1 if u(p) 1,u(p) if 0 < u(p) < 1,

0 if u(p) 0.

Then u

Fand

E(u, u)

E(u, u) for any u

F.

R(E,F)(x, y): the resistance metric on X associated with (E, F)Theorem 1.2. R(E,F)(, ) is a meric on X. For any u F,

|u(x) u(y)|2 R(E,F)(x, y)E(u, u)

for any x, y X.For simplicity, we use R(x, y) instead of R(E,F)(x, y).

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G0 G1 G2

p1

p2 p3

Figure 1: Approximation of the Sierpinski gasket by graphs Gm

Examples of resistance forms

(1) 1-dim. Brownian motion:

E(u, v) =R

du

dx

dv

dxdx

F= {u|E(u, u) < +} = H1(R)R(x, y) = |x y|

(2) Standard resistance form on the Sierpinski gasket: For i = 1, 2, 3,

Fi(z) = (zpi)/2 + piK: the Sierpinski gasket

K = F1(K) F2(K) F3(K)V0 = {p1, p2, p3}

Vm+1 = F1(Vm) F2(Vm) F3(Vm)Em(u, u) = 1

2

(p, q) is an edge of the Graph Gm

53

m(u(p) u(q))2

F= {u| limm

Em(u, u) < +}E(u, v) = lim

mEm(u, v)

(E, F): the standard resistance form on KR(x, y) |x y|(log 5log 3)/ log2

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(3) Random walks on (weighted) graphs (V, C):

V: a countable set,{C(x, y)}x,yV: the conductances, C(x, y) = C(y, x) 0, C(x, x) = 0Assume thatLocally finite: {y|C(x, y) > 0} is finiteConnected(irreducible): For any x, y V, {x1, . . . , xn} such thatx1 = x, xn = y and C(xi, xi+1) > 0 for any i

Random walk associated with (V, C):

C(x) = y

C(x, y): the weight of x

P(x, y) =C(x, y)

C(x): the transition probability from x to y

Pn(x, y) =zV

Pn1(x, z)P(z, y): the transition probability at the time n

Pn(x, y): the heat kernel associated with the random walk

Resistance form associated with (V, C):

F= {u|u : V R,x,y

C(x, y)(u(x) u(y))2

< +}

E(u, v) = 12

x,y

C(x, y)(u(x) u(y))(v(x) v(y)).

(E, F) is a resistance form on V.

Barlow-Coulhon-Kumagai: relations between the heat kernel estimate andthe geometric property of the resistance metric

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Plan: to find a metric d which satisfies (RVD):

Resistance Volume (Distance),

good heat kernel estimate

(This principle is known to work well for other situations as well.)To preserve some desirable properties of the resistance form, we require

d: quasisymmetric with respect to R.

Quasisymmetric maps (QS maps for short): Tukia & Vaisala

a generalization of quasiconformal functions on C

Definition 1.3. (X, d) and (X, ): metric spaces: quasisymmetric, or QS for short, with respect to d

def

a homeomorphism h : [0, ) [0, ) such that h(0) = 0 and

d(x, z) < td(x, y) (x, z) < h(t)(x, y)

We write QS

d.

Fact: QS

d d QS

.

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Regulation of Jumps: Annulus comparable condition

(E, F): a resistance form on X

(E, F) is local def

E(u, v) = 0 if inf{R(x, y)|x supp(u), y supp(v)} > 0

No Jumps

Definition 1.4. (E, F) satisfies the Annulus Comparable Condition,(ACC) for short,

def(X, R) is uniformly perfect and > 0 such that

R(x, BR(x, r)c)

R(x, BR(x, (1 + )r) BR(x, r)

c (1.1)

Annulusfor any x X and any r > 0 with BR(x, r) = X.

R(A, B) =

inf{E(u, u)|u|A 1, u|B 0, u F}1

Local Equality in (1.1) (ACC)

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Volume doubling property

Definition 1.5. (X,d,): a measure metric space is volume doulbing with respect to d, (VD)d for short def c > 0 such that

(Bd(x, 2r)) c(Bd(x, r))for any r > 0 and any x X.(QS) preserves the volume doubling prperty: if d

QS, then

is (VD)d is (VD)

Conclusion: (ACC) and is (VD)R

(ACC) and d : d QS

R, > 0 such that

p(t,x,x) 1Vd(x, t1/)

: the diagonal heat kernel estimate (DHK),

and

p(t,x,x)

Cp(2t,x,x) : the kernel doubling property (KD)

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Resistance forms

Quasisymmetric maps

Heat kernel estimate

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2 Resistance forms

(E, F): a resistance form on a set X

2.1 Topology given by a resistance form

B X. Define

F(B) = {u|u F, u|B 0}BF = {x|x X, u(x) = 0 for any u F(B)}

CF =

{B

|B

X, BF = B

}satisfies the axiom of closed sets.

||F-topology

R-topology: the topology given by the resistance meric R.

Proposition 2.1. (1) F-closed R-closed(2) If(X, R) is compact, then the converse of (1) is also true.(3) In general, the converse of (1) is not true.

Notation.C(X) =

{u|u is continuous with respect to R-topology

}C0(X) = {u|u C(X), supp(u) is R-compact.}Definition 2.2. (E, F) is regular

def

F C0(X) is dense in C0(X) in the sense of ||u|| = supxX |u(x)|.Theorem 2.3. (E, F): regular F-toplogy = R-topologyIn particular, (X, R): compact (E, F): regularHereafter, we always assume that (E, F) is regular.

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2.2 Greens function

Theorem 2.4. B X: closed Unique gB : X X [0, +) with(GF) Define gxB(y) = gB(x, y). Then g

xB F(B). For any u F(B) and

any x X,

E(gxB, u) = u(x)

gB(x, y): the Green function with the boundary Bor the B-Green function

gB(x, x) gB(x, y) 0gB(x, y) = gB(y, x)

gB(x, x) > 0 x / B|gB(x, y) gB(x, z)| RB(y, z) R(y, z)

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Moreover, define

FB = F(B) + R = {u|u F, u is constant on B}XB = (X\B) {B} : shrinking B into a point

Then (E, FB) is a resistance form on XB.RB(, ): associated resistance metric on XB.Then, (due to Metz in case B = {z}),

gB(x, y) =RB(x, B) + RB(y, B) RB(x, y)

2

Gromov product of the metric RB

If B = {z}, then RB(x, y) = R(x, y).

In general,(X, d): a metric space. Define

k(x, y) =d(x, z) + d(y, z) d(x, y)

2

(Au)(x) = X

k(x, y)f(y)(dy)

What is A?

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2.3 Harmonic functions and Traces

B X: closedDefine

F|B = {u|B : u F}.

Proposition 2.5. For any F|B, unique f Fsuch that f|B = and

E(f, f) = minuF,u|B=

E(u, u)

f: the harmonic function with boundary value on the boundary Bor the B-harmonic function with boundary value .

Define f = hB() and HB = hB(F|B). Then

hB : F|B HB F is linear.F= HB F(B) (**)

E(u, v) = 0 ifu HB and v F(B).

In the case of Dirichlet forms, analogous decomposition as (**) is known.

See Fukushima-Oshima-Takeda.Define

E|B(, ) = E(hB(), hB())for any , F|B. ThenProposition 2.6. (E|B, F|B) is a resistance form on B.The corresponding resistance metric = R|BB.(E, F): regular (E|B, F|B): regular(E|B, F|B): the trace of (E, F) on B.

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2.4 Dirichlet form associated with (

E,

F)

Assume that: a Radon measure on (X, R)0 < (BR(x, r)) < + for any x X and any r > 0.Define

E1(u, v) = E(u, v) +X

uvd

D = E1-closure ofF C0(X).

Theorem 2.7.(E, F): regular (E, D): a regular Dirichlet form on L2(X, )Moreover, (E, F): local (E, D): local.

a regular Dirichlet form

a Hunt process, i.e. a strong Markov process with right continuous pathes

local pathes are continuous. (Diffusion)

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Definition 2.8 (Capacity). (1) U

X: open,

CapU = inf{E1(u, u)|u F, u 1 on U}

(2) A X,CapA = inf{CapU|U: open, A U}

Fact For any x X, cx > 0 such that, for any u D,|u(x)| cx

E1(u, u)

K X: compact, 0 < infxKCap{x}.

the Hunt process is determined for all x X.In general, the Hunt process associated with a regular Dirichlet form isdetermined up to excpetionl sets.

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2.5 Transition density/Heat kernel

: a Radon measure on (X, R), 0 < (BR(x, r)) < +(E, F): a regular resistance form on X.

(E, D): a regular Dirichlet form on L2(X, )

({Xt}t>0, {Px}xX): a Hunt process on X

(defined for every x X)Theorem 2.9. Assume that BR(x, r) is compact for any x, X and r > 0.Then there exists p(t,x,y) : (0, ) X X [0, ), continuous with(TD1) pt,x D, where pt,x(y) = p(t,x,y).(TD2) p(t,x,y) = p(t,y,x)(TD3) For any mesurable u 0,

Ex(u(Xt)) =

X

p(t,x,y)u(y)(dy).

(TD4)

p(t + s,x,y) =

X

p(t,x,z)p(s,z,y)(dz)

p(t,x,y): the transition density/heat kernel

Existence and continuity of the transition densityChen et al: general regular Dirichlet forms, ultarcontractive quasicontinuousGrigoryan: general regular Dirichlet, locally ultracontractive quasicontinuousCroydon: resistance forms, ultracontractive continuous

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Proposition 2.10. Without any further assumption,

p(r(BR(x, r)), x , x) 2 +2

(BR(x, r))

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3 Goemetry and analysis on (X, R) via

quasisymmetric maps

3.1 Exit time, resistance and annulus comparablity

Definition 3.1. (X, d): a metric space(X, d): uniformly perfect

def > 0 such that

Bd(x, (1 + )r)\Bd(x, r) = for any x X and r > 0 with X\Bd(x, r) = .Hereafter, (E, F): a regular resistance form on X

: a randon measure on (X, R)

BR(x, r): compact(E, D) a regular Dirichlet form on L2(X, )

({Xt}t>0, {Px}x>0): regular Hunt processp(t,x,y): the transition density

For simplicity, we only give statements the case where (X, R) is notbounded.

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Recall the Annulus comparable condition (ACC):

> 0 such that

R(x, BR(x, r)c) R(x, BR(x, (1 + )r) BR(x, r)c).

Definition 3.2 (Exit time). A X,

A = inf{t > 0|Xt / A}.

Proposition 3.3.

Ex(A) =X

gAc(x, y)(dy) =A

RAc(x, Ac) + RAc(y, A

c)

RAc(x, y)

2 (dy)

Theorem 3.4. Assume: (VD)R, i.e. volume doublig with repsect to R,(X, R): uniformly perfectd: a metric on X, d

QSR, i.e. d is quasisymmetric with respect to R.

Then(ACC)

Exit time estimate (Exit)

d

: Ex(Bd

(x,r))

Rd(x, r)Vd(x, r)

Resistance estimate (Res)d: R(x, Bd(x, r)

c) Rd(x, r),

where Rd(x, r) = supyBd(x,r) R(x, y).

Exit time estimate: Resistance Volume Exit time

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Assume that (X, R) is uniformly perfect.

Theorem 3.5. If d QS

R, (ACC) holds, : (VD)R, then

p(Rd(x, r)Vd(x, r), x , x) 1Vd(x, r)

: Diagonal estimate

and

p(Rd(x, r)Vd(x, r), x , y) cVd(x, r)

: Near diagonal lower estimate

for x, y

X with d(x, y)

cr.

In particular, if d = R, then

p(rVR(x, r), x , x) 1VR(x, r)

X is a graph, random walk, d = R: Barlow-Coulhon-Kumagaid = R, continuous: Kumagai

Observation: In the diagonal estimate, if Rd(x, r)Vd(x, r)

r, then


.

Find d QS

R such that Resistance Volume = (Distance) : (RVD)!!

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3.2 Construction of quasisymmetric metric

(X,,): a metric measure spaceAssume that (X, ) is uniformly perfect.

Theorem 3.6. Fix a 0.If : (VD) then, for sufficiently large > 0, d: a metric on X such that

QSd and

(x, y)aVd(x, d(x, y)) d(x, y) (M)

(M) is a natural analogue of (RVD).

Remark. In the case a = 0, the above theorem recovers the followingfamous result:If (X, ) is uniformly perfect and is (VD), then there exists a metric d onX such that d

QS and is Ahlfors regular, i.e.

(Bd(x, r)) r

For > 0, define the condition (SD): slow decay of volume

: (0, 1]

(0,

), ()

0 as

0 monotonically, and, for any

(0, 1],

any x, y X,Vd(x,d(x, y))

Vd(x, d(x, y))

()

Theorem 3.7. Fix a > 0. Assume that (X, d) is uniformly perfect. Then(SD) (M)

QSd (M)

is (VD) and (VD)d.

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4 Heat kernel estimate

4.1 Main Theorems

List of Conditions:

(DHK)d,: the diagonal heat kernel estimate


(KD): kernel doubling, c > 0,

p(t,x,x) cp(2t,x,x)

(RVD)d,: Resistance Volume = Distance

R(x, y)Vd(x, d(x, y)) d(x, y)

(SD)d,: slow decay of volume : (0, 1] (0, +), () 0 as 0 monotonically and

Vd(x,d(x, y))

Vd(x, d(x, y))

()

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Theorem 4.1. Assume that (X, R) is uniformly perfect. Then

(X, d): uniformly perfect (SD)d, (RVD)d,

d QS

R (RVD)d,

d QS

R (DHK)d, (KD)

Moreover, if (E, F) is local, then the above set of conditioins implies

p(t,x,y) c1Vd(x, t1/)

exp

c2

d(x, y)

t

1/(1))

If (E, F) is local and d is geodesic, then

c3Vd(x, t1/)

exp

c4

d(x, y)

t

1/(1)) p(t,x,y)

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Theorem 4.2. Assume (X, R) is uniformly perfect. Then

: (VD)R (ACC)

: (VD)R R(x, BR(x, r)c) r

(ACC) d and > 0 such that d QS

R, (DHK)d, and (KD)

Remark. local (ACC) and/or R(x, BR(x, r)c) r

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4.2 Applicaition to traces

Assume that (X, R) is uniformly perfect.

B X: closedConsider the trace (E|B, F|B) of (E, F) on B.Recall that

(E, F): regular (E|B, F|B): regularTheorem 4.3. Assume that (B, R|B) is uniformly perfect.

(ACC) for (E, F) (ACC) for (E|B, F|B).

Assumptions:(X, R) and (B, R|B): uniformly perfect(E, F): regular(ACC) holds for (E, F).BR(x, r): compact

: a Radon measure on (B, R|B)

(E|B, DB): a regular Dirichlet form on L2(B, ).

Transition density: pB

(t,x,y) on BTheorem 4.4. Assume that d

QSR and (DHK)d,.

If > 0 such that

(Bd(x, r)) r(Bd(x, r) B),

then > and

pB (t,x,x) 1

(Bd(x, t1/()) B)Moreover, if (Bd(x, r))

r, then

pB (t,x,x) t .

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4.3 Examples

-stable process on R1: (1, 2]

E()(u, v) =R2

|u(x) u(y)|2|x y|1+ dx

F() = {u|u C(R), E()(u, u) < +}R()(x, y) = c|x y|1

for (1, 2). For = 2, it corresponds to the Brownian motion on R1.(ACC) is OK.

Case 1: = dx the Lebesgue measure. Then is (VD)R.

p(t,x,x) 1t1/

.

Case 2: = xdx for > 1 is (VD)R.

p(t, 0, 0) t : = + 1+

Case 3: Trace onto the middle 3rd Cantor set K:

: the log3/ log 2-dim. Hausdorff measure on K. Let be the Lebesguemeasure.

(BR(x, r)) rlog 2

(1)log 3 (BR(x, r))

pK (t,x,x) t : =log2

( 1) log 3 + log 2

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The standard resistance form on the Sierpinski gasket

Natural measure = the log 3/ log 2-dim. Hausdorff measure.

p(t,x,y) c1t/

exp

c2

d(x, y)

t

1/(1)),

where =log3

log2, =

log5

log2and d(x, y) = |x y| = R(x, y) log 2log 5log 2 .

Case 1: Change the measure :

Case 2: Trace onto an Ahlfors -regular set B: on Y such that

(Bd(x, r) B) rThen

pB (t,x,x) t : =log2

log5 log 3 + log2

In particular, B = the line segment of the outer triangle: = 1Characterization ofF|B as a Besov space: Alf Jonsson

=log5 log 3 + log 2

log2

F|B = F() = the domain for the -stable process on R.E|B(u, u) E()(u, u)

But...................

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vd rf lecture

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